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BOOK REVIEW

Genius Unappreciated

Brian Hayes

János Bolyai, Non-Euclidean Geometry, and the Nature of Space. Jeremy J. Gray. x + 185 + 54 pp. Burndy Library Publications (distributed by the MIT Press), 2004. $20, paper.

János Bolyai, a Hungarian engineer and army officer, was one of at least three inventors of non-Euclidean geometry. Worried that rivals might overtake him, he hastily published his work in 1832 as an appendix to a book by his father. It was already too late; Nikolai Lobachevskii had published similar ideas in 1829. But Bolyai and Lobachevskii were to share a fate even more bitter than being scooped: They were ignored. Each of them had resolved a question that had been nagging at mathematicians for more than 2,000 years, yet no one paid attention.

János BolyaiClick to Enlarge Image

For Bolyai, this handsome new book from the Burndy Library brings belated vindication. It reproduces the 1832 appendix both in the original Latin and in an English translation by George Bruce Halsted. Perhaps more important, it includes a thoughtful introductory essay and tutorial by Jeremy J. Gray, illustrated with many helpful diagrams. The book, published solely in a paperback edition, is exceptionally well designed, printed and bound.

What's non-Euclidean about Bolyai's geometry? Euclid began his work with five postulates, or statements to be accepted without proof; four of the postulates seemed innocuous, but many readers found the fifth one suspect. A modern statement of this fifth postulate goes like this: In the plane defined by a line L and a point P not on L, exactly one line can be drawn through P that never intersects L. Most skeptics of this parallel postulate didn't doubt its truth but rather its necessity; they thought it could be derived from simpler notions. Bolyai showed something quite different: He constructed a new geometry in which many lines through P are parallel to L. It is the geometry of a strange place—for example, the angles of a triangle sum to less than 180 degrees, and the Pythagorean theorem fails—but Bolyai showed it is internally consistent and just as logical as Euclid's formulation. (Since Einstein introduced general relativity, the strange place described by non-Euclidean geometry has been recognized as our own universe.)

Bolyai's story is a sad one of genius unappreciated. His appendix apparently had only one competent reader during his lifetime, and even that led only to more anguish. The reader was Carl Friedrich Gauss, the most eminent mathematician of the age, who was sent a copy of the book with the appendix. Gauss wrote back, "If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself." He went on to explain that he had entertained the same ideas for 30 or 35 years but had refrained from publishing them. Bolyai was appalled and suspected dishonesty, but a later examination of Gauss's notebooks confirmed that Gauss was indeed the third—or rather the first—inventor of non-Euclidean geometry. Perhaps it's just as well for Bolyai that he died before the notebooks were opened.—Brian Hayes


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